Let F: R3->R3 be the linear transformation defined by the orthogonal projection of (v belongs to R3) onto the subspace W= {(x,y,z)|x+y+z=0}
a. Find the standard matrix A of F
b. Show that A(A - I) = 0 Why would this be true in general (for any subspace W)?

thanks in advance :D

Sep 1st 2009, 01:26 PM

Opalg

Quote:

Originally Posted by pdnhan

Let F: R3->R3 be the linear transformation defined by the orthogonal projection of (v belongs to R3) onto the subspace W= {(x,y,z)|x+y+z=0}
a. Find the standard matrix A of F
b. Show that A(A - I) = 0 Why would this be true in general (for any subspace W)?

The vector normal to W is . Given a vector in , the effect of F on will be to add a multiple of to so as to take it to the subspace W. The condition for to belong to W is . Solve that to see that . From that, you should be able to write down the matrix A and verify that .

For the last part, if A is the matrix of the orthogonal projection onto a subspace, then I–A is the matrix of the orthogonal projection onto the (orthogonal) complementary subspace.

Sep 2nd 2009, 12:36 AM

pdnhan

cheers man

Sep 2nd 2009, 03:28 PM

pdnhan

hey man, can you please tell me how to get the value of A so I can compare, and your solution to part b) as well?

Sep 3rd 2009, 01:32 AM

Opalg

Quote:

Originally Posted by pdnhan

hey man, can you please tell me how to get the value of A so I can compare, and your solution to part b) as well?

You tell us yours first, then I'll let you know whether I agree. (Evilgrin)